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Find the sum of the expression

$\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+............+\frac{1}{\sqrt{80}+\sqrt{81}}$

1. $7$
2. $8$
3. $9$
4. $10$
edited | 1.2k views

when you such overlapping expressions just rationalise it and add in most of the case you will be left with lesser number of terms ..in this case i am left with $\sqrt{81}-\sqrt{1}=8.$

$\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{4}}+..........+\dfrac{1}{\sqrt{80}+\sqrt{81}}$

$=\dfrac{1}{\sqrt{1}+\sqrt{2}}\times\left(\dfrac{\sqrt{1}-\sqrt{2}}{\sqrt{1}-\sqrt{2}} \right )+\dfrac{1}{\sqrt{2}+\sqrt{3}}\times\left(\dfrac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}} \right )+\dfrac{1}{\sqrt{3}+\sqrt{4}}\times\left(\dfrac{\sqrt{3}-\sqrt{4}}{\sqrt{3}-\sqrt{4}} \right )+.........+\dfrac{1}{\sqrt{80}+\sqrt{81}}\times\left(\dfrac{\sqrt{80}-\sqrt{81}}{\sqrt{80}-\sqrt{81}} \right )$

$=\dfrac{\sqrt{1}-\sqrt{2}}{(\sqrt{1})^{2}-(\sqrt{2})^{2}}+\dfrac{\sqrt{2}-\sqrt{3}}{(\sqrt{2})^{2}-(\sqrt{3})^{2}}+\dfrac{\sqrt{3}-\sqrt{4}}{(\sqrt{3})^{2}-(\sqrt{4})^{2}}+..........+\dfrac{\sqrt{80}-\sqrt{81}}{(\sqrt{80})^{2}-(\sqrt{81})^{2}}$

$=-\left(\sqrt{1}-\sqrt{2}+\sqrt{2}-\sqrt{3} +\sqrt{3}-\sqrt{4}+.........+\sqrt{80}-\sqrt{81}\right )$

$=\sqrt{81}-\sqrt{1}$

$= 8$
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Simplest Approach!

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